Theorem nzrnz

Description: One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015)

	Ref	Expression
Hypotheses	isnzr.o	$$⊢\underset{\dot{}}{1}=1_R$$
	isnzr.z	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	nzrnz	$$⊢R\in \mathrm{NzRing}\to \underset{\dot{}}{1}\ne \underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		isnzr.o	$$⊢\underset{\dot{}}{1}=1_R$$
2		isnzr.z	$$⊢\underset{\dot{}}{0}=0_R$$
3	1 2	isnzr	$$⊢R\in \mathrm{NzRing}\leftrightarrow \left(R\in \mathrm{Ring}\wedge \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\right)$$
4	3	simprbi	$$⊢R\in \mathrm{NzRing}\to \underset{\dot{}}{1}\ne \underset{\dot{}}{0}$$